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Suspensions are composed of mixtures of particles and fluid and are omnipresent in natural phenomena and in industrial processes. The present paper addresses the rheology of concentrated suspensions of non-colloidal particles. While hydrodynamic interactions or lubrication forces between the particles are important in the dilute regime, they become of lesser significance when the concentration is increased, and direct particle contacts become dominant in the rheological response of concentrated suspensions, particularly those close to the maximum volume fraction where the suspension ceases to flow. The rheology of these dense suspensions can be approached via a diversity of approaches that the paper introduces successively. The mixture of particles and fluid can be seen as a fluid with effective rheological properties but also as a two-phase system wherein the fluid and particles can experience relative motion. Rheometry can be undertaken at an imposed volume fraction but also at imposed values of particle normal stress, which is particularly suited to yield examination of the rheology close to the jamming transition. The response of suspensions to unsteady or transient flows provides access to different features of the suspension rheology. Finally, beyond the problem of suspension of rigid, non-colloidal spheres in a Newtonian fluid, there are a great variety of complex mixtures of particles and fluid that remain relatively unexplored.

The pinch-off of a capillary thread is studied at large Ohnesorge number for non-Brownian, neutrally buoyant, mono-disperse, rigid, spherical particles suspended in a Newtonian liquid with viscosity
$\unicode[STIX]{x1D702}_{0}$
and surface tension
$\unicode[STIX]{x1D70E}$
. Reproducible pinch-off dynamics is obtained by letting a drop coalesce with a bath. The bridge shape and time evolution of the neck diameter,
$h_{\mathit{min}}$
, are studied for varied particle size
$d$
, volume fraction
$\unicode[STIX]{x1D719}$
and liquid contact angle
$\unicode[STIX]{x1D703}$
. Two successive regimes are identified: (i) a first effective-viscous-fluid regime which only depends upon
$\unicode[STIX]{x1D719}$
and (ii) a subsequent discrete regime, depending both on
$d$
and
$\unicode[STIX]{x1D719}$
, in which the thinning localises at the neck and accelerates continuously. In the first regime, the suspension behaves as an effective viscous fluid and the dynamics is solely characterised by the effective viscosity of the suspension,
$\unicode[STIX]{x1D702}_{e}\sim -\unicode[STIX]{x1D70E}/{\dot{h}}_{\mathit{min}}$
, which agrees closely with the steady shear viscosity measured in a conventional rheometer and diverges as
$(\unicode[STIX]{x1D719}_{c}-\unicode[STIX]{x1D719})^{-2}$
at the same critical particle volume fraction,
$\unicode[STIX]{x1D719}_{c}$
. For
$\unicode[STIX]{x1D719}\gtrsim 35\,\%$
, the thinning rate is found to increase by a factor of order one when the flow becomes purely extensional, suggesting non-Newtonian effects. The discrete regime is observed from a transition neck diameter,
$h_{\mathit{min}}\equiv h^{\ast }\sim d\,(\unicode[STIX]{x1D719}_{c}-\unicode[STIX]{x1D719})^{-1/3}$
, down to
$h_{\mathit{min}}\approx d$
, where the thinning rate recovers the value obtained for the pure interstitial fluid,
$\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D702}_{0}$
, and lasts
$t^{\ast }\sim \unicode[STIX]{x1D702}_{e}h^{\ast }/\unicode[STIX]{x1D70E}$
.

Pressure- and volume-imposed rheology is used to study suspensions of non-colloidal, rigid fibres in the concentrated regime for aspect ratios ranging from 3 to 15. The suspensions exhibit yield stresses. Subtracting these apparent yield stresses reveals a viscous scaling for both the shear and normal stresses. The variation in aspect ratio does not affect the friction coefficient (ratio of shear and normal stresses), but increasing the aspect ratio lowers the maximum volume fraction at which the suspension flows. Constitutive laws are proposed for the viscosities and the friction coefficient close to the jamming transition.

The large-amplitude oscillatory flow of a suspension of spherical particles in a pipe is studied at low Reynolds number. Particle volume fraction and velocity are examined through refractive index matching techniques. The particles migrate toward the centre of the pipe, i.e. toward regions of lower shear rate, for bulk volume fractions larger than 10 %. Steady results are in agreement with available experimental results and discrete-particle simulations for similar geometries. The dynamics of the shear-induced migration process are analysed and compared against the predictions of the suspension balance model using realistic rheological laws.

Pressure-imposed rheometry is used to study the rheological properties of suspensions of non-colloidal spheres in yield-stress fluids. Accurate measurements for both the shear stress and the particle normal stress are obtained in the dense regime. The rheological measurements are favourably compared with a model based on scaling arguments and homogenisation methods.

Measurements of normal stress differences are reported for suspensions of rigid, non-Brownian fibres for concentrations of
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}nL^2d=1.5\text {--}3$
and aspect ratios of
$L/d=11\text {--}32$
, where
$n$
is the number of fibres per unit volume,
$L$
is the fibre length and
$d$
is the diameter. The first and second normal stress differences are determined experimentally from measuring the deformation in the free surface in a tilted trough and in a Weissenberg rheometer. Simulations are performed as well, and the hydrodynamic and contact contributions to the normal stresses are calculated. The experiments and simulations indicate that the second normal stress difference is negative and that its magnitude increases as the concentration is raised and the aspect ratio is lowered. The first normal stress difference is positive and its magnitude is approximately twice that of the second normal stress difference. Simulation results indicate that, for the concentrations and aspect ratios studied, contact forces between fibres form the dominant contribution to the normal stress differences.

The mobile layer of a granular bed composed of spherical particles is experimentally investigated in a laminar rectangular channel flow. Both particle and fluid velocity profiles are obtained using particle image velocimetry for different index-matched combinations of particles and fluid and for a wide range of fluid flow rates above incipient motion. A full three-dimensional investigation of the flow field inside the mobile layer is also provided. These experimental observations are compared to the predictions of a two-phase continuum model having a frictional rheology to describe particle–particle interactions. Different rheological constitutive laws having increasing degrees of sophistication are tested and discussed.

The purpose of this book is to provide an introduction to suspension dynamics, and so we (the authors) thought it would be good to give some historical (as well as personal) perspective on the study of suspensions. Early development of the subject was largely due to two “schools,” one in England and one in the United States. In England, the subject developed from the fluid mechanical tradition at the University of Cambridge, dating from the work of G. G. Stokes and H. Lamb in the mid- and late-1800s. The subject developed in earnest from the work of George Batchelor and collaborators at Cambridge's Department of Applied Mathematics and Theoretical Physics (DAMTP). In the United States, the development of the discipline took place primarily in chemical engineering departments, largely through the efforts of Andreas Acrivos and a number of his students at the University of California Berkeley, Stanford University, and the City College of New York (CCNY). The authors' approaches to suspensions owe much to these “schools” of suspension dynamics. Élisabeth Guazzelli was introduced to the subject by Bud Homsy at Stanford University and extended interactions with John Hinch of the University of Cambridge. Jeff Morris received his introduction to suspensions as a doctoral student of John Brady (a student of Acrivos) at the California Institute of Technology.

Understanding the behaviour of particles suspended in a fluid has many important applications across a range of fields, including engineering and geophysics. Comprising two main parts, this book begins with the well-developed theory of particles in viscous fluids, i.e. microhydrodynamics, particularly for single- and pair-body dynamics. Part II considers many-body dynamics, covering shear flows and sedimentation, bulk flow properties and collective phenomena. An interlude between the two parts provides the basic statistical techniques needed to employ the results of the first (microscopic) in the second (macroscopic). The authors introduce theoretical, mathematical concepts through concrete examples, making the material accessible to non-mathematicians. They also include some of the many open questions in the field to encourage further study. Consequently, this is an ideal introduction for students and researchers from other disciplines who are approaching suspension dynamics for the first time.

In this chapter, our primary purpose is to go beyond Stokes flow to tackle the very difficult problem of understanding the influence of fluid inertia on particle-laden flows. Specifically, the issue of interest is the effect of inertia at the particle scale. Following the structure of the preceding two chapters, we consider first the influence of inertia on sedimentation, and then on shear flows of particle-laden fluid, where we will also consider the rheological consequences of inertia. Inclusion of inertia changes the form of the equation of motion, and even weak inertia can have singular effects when large domains are considered; for both sedimentation and shear, we provide a sketch of results obtained using the singular perturbation method of matched asymptotic expansions in the limit of weak inertia, i.e. at small Reynolds number. While Stokes flow is a good approximation near the particle, a pronounced change in symmetry of the disturbance flow caused by the particle is seen if we are far enough away, as the fore–aft symmetry of Stokes flow is completely lost in this “far-field” region.

We can only give an outline of the subject of inertial suspension flow, as most issues are far from completely resolved. In the previous two chapters, the issues which remain unclear are primarily collective, whereas the microhydrodynamic theory is well-established. For inertial suspensions, the level of understanding at the microscopic, i.e. single and pair, level is incomplete. Hence understanding of collective phenomena based on the microscopic physics is not well-developed and may expand rapidly.

In this chapter, we describe shear flows of suspensions. The goal here is to illustrate the connection between the particle-scale interactions and bulk suspension phenomena. At the microscopic scale, we consider the interactions of discrete particles and the resulting microstructural arrangement, while at the bulk scale the mixture is described as a continuous effective fluid. The connection between the scales is provided by the rheology, i.e. by the stress response of the bulk material. This chapter describes non-Newtonian properties as well as shear-induced diffusivity exhibited by suspensions, and presents an introduction to the relationship between these properties and the flow-induced microstructure. Irreversibility of the bulk motion seen in shear-induced particle migration demonstrates how the interplay of Stokes-flow hydrodynamics, outlined in Part I, with other particle-scale forces leads to some unexpected behavior. As we consider the average material behavior and its relation to the microscopic interactions, it is natural to apply concepts from statistical physics introduced in Chapter 5.

A number of the issues raised in this chapter are topics of active research in rheology and multiphase flow; while we provide a few references as a guide to further information on specific issues, recent reviews by Stickel and Powell (2005), Morris (2009), and Wagner and Brady (2009) provide fuller coverage of the literature.

Suspension viscosity

We have all heard that “blood is thicker than water.” Blood is, in fact, a suspension of red blood cells in a Newtonian plasma.

Finally in this book, we would like to broaden the discussion to topics where the understanding is less clear. The future of the subject will involve study of these open questions, but we do not intend to suggest that the list of topics that we are discussing is all-inclusive, or even to suggest these topics as priorities. Instead we seek to provide some indication of the scope of activities for which the concepts developed in this book may find future use.

Moving toward open questions While some of the issues discussed in the last chapters are mostly settled (or perhaps will be resolved soon), there remain greater challenges in many areas of suspension flows. We are perhaps touching on the more obvious of issues which come to mind following the exposition in the preceding chapters, and thus we likely miss novel avenues of study. Nonetheless, a list of issues in suspensions where many open questions remain includes:

Dense suspensions: Flow of suspensions approaching the maximum packing limit is often referred to as “dense suspension flow” and this condition raises special issues which we have only noted briefly in this book. In particular, for such mixtures, the particle surfaces are likely to make enduring contacts, and the details of surface roughness and friction coefficient will play a role in the behavior. How such contact forces interact with hydrodynamic lubrication forces in dense suspensions, and the relation of dense suspension flow to dense granular flow in which the interstitial fluid is a gas, are open questions of interest.

In Part I, we have presented the basis of microhydrodynamics. In its broad definition, microhydrodynamics represents the theory of viscous fluid flows at small spatial scales. For the purposes of the present book on suspensions, we have specifically considered the single- and pair-body dynamics of small particles immersed in viscous fluid. In Part II, we will describe macroscopic phenomena encountered in flows involving a large number of particles interacting through viscous fluid. We will also provide an introduction to methods developed for understanding and (hopefully) predicting certain macroscopic phenomena in suspensions in terms of the microscopic concepts described in Part I, combined with ideas that fall in the realms of statistical physics and dynamical systems.

In this transitional chapter, we are concerned primarily with introducing statistical techniques and concepts from stochastic processes which we will apply in the following chapters. We will also briefly consider the related issue of chaotic dynamics.

Statistical physics

The theoretical framework for relating microscopic mechanics to macroscopic or bulk properties is statistical physics or statistical mechanics. Understanding systems made up of many interacting particles is far from trivial, and the difficulty involved is not just a mere question of solving the hydrodynamic equations with better, faster computers. The collective interactions between the particles can give rise to quite unexpected qualitative behavior, often much simpler than the microscopic motions seem to suggest.

Mobile particulate systems are encountered in various natural and industrial processes. In the broadest sense, mobile particulate systems include both suspensions and granular media. Suspensions refer to particles dispersed in a liquid or a gas. Familiar examples include aerosols such as sprays, mists, coal dust, and particulate air pollution; biological fluids such as blood; industrial fluids such as paints, ink, or emulsions in food or cosmetics. Suspension flows are also involved in numerous material processing applications, including manufacture of fiber composites and paper, and in natural processes such as sediment transport in rivers and oceans. In common usage, a suspension refers to solid particles as the dispersed state in a liquid, while an emulsion concerns liquid droplets dispersed in another immiscible fluid, and an aerosol is specific to the case of a suspension of fine solid or liquid particles in a gas. We focus on the case of a suspension in this text.

In the flow of suspensions, the viscous fluid between the particles mediates particle interactions, whereas in dry granular media the fluid between the particles is typically assumed to have a minor role, doing no more than providing a resistive drag, and this allows direct contact interactions. Familiar examples of granular media include dry powders, grains, and pills in the food, pharmaceutical, and agricultural industries; sand piles, dredging, and liquefaction of soil in civil engineering; and geophysical phenomena such as landslides, avalanches, and volcanic eruptions.